In physics papers, the massless free boson has a definition involving an action:

$$ S(X) = \frac{1}{8\pi} \int d\sigma^2\,   \partial X \overline{\partial X}$$

The random functions $X(z)$ are sampled according to the Gaussian distribution: $e^{-S(X)}$.  In this case we can define the correlator to be:

$$ \mathbb{E}[X(z,\overline{z})X(w, \overline{w})] = - \ln |z - w|^2$$

See Ginsparg, applied [Conformal Field Theory](http://arxiv.org/abs/hep-th/9108028). I am trying to fill in the details of this derivation... using probability theory, complex analysis or anything else.

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Since $X$ is a [bosonic](https://en.wikipedia.org/wiki/Boson) field I will just model it as a function... judging from the norm it should live in some hardy space $X \in H^2$ - the complexification of $L^2$ - where we should fill in some Riemann surface with flat structure. I think Ginsparg uses $H^2(\mathbb{R}\times S^1)$.

Then I would like to expand.  $\partial X(z) = \sum a_n z_n$ and get

$$ \langle \partial X(z)\partial X(w) \rangle = \sum \mathbb{E}[a_n^2]z \overline{w}  = \frac{1}{|z-w|^2}$$

There are holes everywhere in this argument - a real physicists's derivation.  How can I make this more rigorous?  Especially the intuition about the Gaussian random function in $L^2$ ?

Thanks.